There exists several results connecting pure LDP to the contraction properties of KL divergence \(D_{\mathsf {KL}}\)  and total variation distance \({\mathsf {TV}}\) . For instance, for any \(\varepsilon \) -LDP mechanism \({\mathsf {K}}\) , it is shown in [1]} that \(D_{\mathsf {KL}}(P{\mathsf {K}}\Vert Q{\mathsf {K}})\le 2(e^\varepsilon -1)^2{\mathsf {TV}}^2(P, Q)\)  and in [2]} that \({\mathsf {TV}}(P{\mathsf {K}}\Vert Q{\mathsf {K}})\le \frac{e^\varepsilon -1}{e^\varepsilon + 1}{\mathsf {TV}}(P, Q)\)  for any pairs \((P, Q)\) .
Inspired by these results, we further show that if \({\mathsf {K}}\)  is \((\varepsilon , \delta )\) -LDP then \(D_f(P{\mathsf {K}}\Vert Q{\mathsf {K}})\le [1-e^{-\varepsilon }(1-\delta )] D_f(P\Vert Q)\)  for any arbitrary \(f\) -divergences \(D_f\)  and any pairs \((P, Q)\) .
